A very successful quantum system that has been investigated to date is a two-state system (TSS) coupled to a dissipative environment. Despite its simplicity, the TSS dissipative dynamics is the paradigm of a wide variety of physical systems. Superconducting devices containing Josephson junctions, few-electron semiconductor quantum dots and two-level atoms in optical cavities are just few examples of systems whose low-energy level dynamics can be, in general, captured by a dissipative TSS.

An emerging problem in the area of dissipative TSSs is the one in which the bath effective spectral density Jeff(ω) presents a pronounced peak (resonance) at a characteristic frequency Ω. A typical example of such a case happens when the energy scale of the device used to detect the state of the TSS is comparable to that of its own regime of operation. Since the device, or 'quantum detector', also suffers from the dissipative effects of the environment, the TSS- detector resonance constitutes an efficient channel for decoherence processes to take place.

We propose an approximation scheme to describe the dynamics of the spin-boson model when the spectral density of the environment shows a peak at a characteristic frequency Ω which can be very close (or even equal) to the spin Zeeman frequency Δ. Mapping the problem onto a TSS coupled to a harmonic oscillator (HO) with frequency ω0, we show that the representation of displaced HO states provides an appropriate basis to truncate the Hilbert space of the TSS-HO system and therefore a better picture of the system dynamics. We derive an effective Hamiltonian for the TSS-HO system, and show it furnishes a very good approximation for the system dynamics even when its two subsystems are moderately coupled. Finally, assuming the regime of weak HO-bath coupling and low temperatures, we are able to analytically evaluate the dissipative TSS dynamics.